A lopsided scaled DTS preconditioning method for the discrete space-fractional diffusion equations

The Grünwald–Letnikov type formulas are applied to discretize the space-fractional diffusion equation and the discretization result is got as a system of linear equations. A lopsided scaled diagonal and Toeplitz splitting (LSDTS) iteration method is constructed to solve this linear system and the convergence is also discussed. An LSDTSτ preconditioner is proposed based on the LSDTS iteration method and the GMRES method combined with this preconditioner is applied to solve the linear system. We theoretically show that the eigenvalues of the LSDTSτ preconditioned matrix are clustered. The numerical experiments illustrate that the LSDTSτ preconditioner can significantly improve the convergence properties of the GMRES method.